Analysis of multilevel methods for eddy current problems

In papers by Arnold, Falk, and Winther, and by Hiptmair, novel multigrid methods for discrete -elliptic boundary value problems have been proposed. Such problems frequently occur in computational electromagnetism, particularly in the context of eddy current simulation.

This paper focuses on the analysis of those nodal multilevel decompositions of the spaces of edge finite elements that form the foundation of the multigrid methods. It provides a significant extension of the existing theory to the case of locally vanishing coefficients and nonconvex domains. In particular, asymptotically uniform convergence of the multigrid method with respect to the number of refinement levels can be established under assumptions that are satisfied in realistic settings for eddy current problems.

The principal idea is to use approximate Helmholtz-decompositions of the function space into an -regular subspace and gradients. The main results of standard multilevel theory for -elliptic problems can then be applied to both subspaces. This yields preliminary decompositions still outside the edge element spaces. Judicious alterations can cure this.

     

Canonical construction of finite elements

The mixed variational formulation of many elliptic boundary value problems involves vector valued function spaces, like, in three dimensions, and . Thus finite element subspaces of these function spaces are indispensable for effective finite element discretization schemes. Given a simplicial triangulation of the computational domain , among others, Raviart, Thomas and Nédélec have found suitable conforming finite elements for and . At first glance, it is hard to detect a common guiding principle behind these approaches. We take a fresh look at the construction of the finite spaces, viewing them from the angle of differential forms. This is motivated by the well-known relationships between differential forms and differential operators: , and can all be regarded as special incarnations of the exterior derivative of a differential form. Moreover, in the realm of differential forms most concepts are basically dimension-independent. Thus, we arrive at a fairly canonical procedure to construct conforming finite element subspaces of function spaces related to differential forms. In any dimension we can give a simple characterization of the local polynomial spaces and degrees of freedom underlying the definition of the finite element spaces. With unprecedented ease we can recover the familiar - and -conforming finite elements, and establish the unisolvence of degrees of freedom. In addition, the use of differential forms makes it possible to establish crucial algebraic properties of the canonical interpolation operators and representation theorems in a single sweep for all kinds of spaces.

     

The toric -vectors of partially ordered sets

An explicit formula for the toric -vector of an Eulerian poset in terms of the -index is developed using coalgebra techniques. The same techniques produce a formula in terms of the flag -vector. For this, another proof based on Fine's algorithm and lattice-path counts is given. As a consequence, it is shown that the Kalai relation on dual posets, , is the only equation relating the -vectors of posets and their duals. A result on the -vectors of oriented matroids is given. A simple formula for the -index in terms of the flag -vector is derived.

     

A computational approach to Hilbert modular group fixed points

Some useful information is known about the fundamental domain for certain Hilbert modular groups. The six nonequivalent points with nontrivial isotropy in the fundamental domains under the action of the modular group for , , and have been determined previously by Gundlach. In finding these points, use was made of the exact size of the isotropy groups. Here we show that the fixed points and the isotropy groups can be found without such knowledge by use of a computer scan. We consider the cases and . A computer algebra system and a C compiler were essential in perfoming the computations.

     

First-hit analysis of algorithms for computing quadratic irregularity

The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta function at odd negative integers, are uniformly distributed modulo for every . This is the basis of a well-known heuristic, given by Siegel, estimating the frequency of irregular primes. So far, analyses have shown that if is a real quadratic field, then the values of the zeta function at negative odd integers are also distributed as expected modulo for any . We use this heuristic to predict the computational time required to find quadratic analogues of irregular primes with a given order of magnitude. We also discuss alternative ways of collecting large amounts of data to test the heuristic.

     

Computational estimation of the constant

The paper describes a computational estimation of the constant characterizing the bounds of . It is known that as

with , while the truth of the Riemann hypothesis would also imply that . In the range , two sets of estimates of are computed, one for increasingly small minima and another for increasingly large maxima of . As increases, the estimates in the first set rapidly fall below and gradually reach values slightly below , while the estimates in the second set rapidly exceed and gradually reach values slightly above . The obtained numerical results are discussed and compared to the implications of recent theoretical work of Granville and Soundararajan.

     

Data-sparse approximation to the operator-valued functions of elliptic operator

In previous papers the arithmetic of hierarchical matrices has been described, which allows us to compute the inverse, for instance, of finite element stiffness matrices discretising an elliptic operator The required computing time is up to logarithmic factors linear in the dimension of the matrix. In particular, this technique can be used for the computation of the discrete analogue of a resolvent

In the present paper, we consider various operator functions, the operator exponential negative fractional powers , the cosine operator function and, finally, the solution operator of the Lyapunov equation. Using the Dunford-Cauchy representation, we get integrals which can be discretised by a quadrature formula which involves the resolvents mentioned above. We give error estimates which are partly exponentially, partly polynomially decreasing.

     

The generalized triangular decomposition

Given a complex matrix , we consider the decomposition , where is upper triangular and and have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where is an upper triangular matrix with the eigenvalues of on the diagonal. We show that any diagonal for can be achieved that satisfies Weyl's multiplicative majorization conditions:

where is the rank of , is the -th largest singular value of , and is the -th largest (in magnitude) diagonal element of . Given a vector which satisfies Weyl's conditions, we call the decomposition , where is upper triangular with prescribed diagonal , the generalized triangular decomposition (GTD). A direct (nonrecursive) algorithm is developed for computing the GTD. This algorithm starts with the SVD and applies a series of permutations and Givens rotations to obtain the GTD. The numerical stability of the GTD update step is established. The GTD can be used to optimize the power utilization of a communication channel, while taking into account quality of service requirements for subchannels. Another application of the GTD is to inverse eigenvalue problems where the goal is to construct matrices with prescribed eigenvalues and singular values.

     

Heuristics for class numbers and lambda invariants

Let be an imaginary quadratic field and let be the associated real quadratic field. Starting from the Cohen-Lenstra heuristics and Scholz's theorem, we make predictions for the behaviors of the 3-parts of the class groups of these two fields as varies. We deduce heuristic predictions for the behavior of the Iwasawa -invariant for the cyclotomic -extension of and test them computationally.

     

Sign changes in sums of the Liouville function

The Liouville function is the completely multiplicative function whose value is at each prime. We develop some algorithms for computing the sum , and use these methods to determine the smallest positive integer where . This answers a question originating in some work of Turán, who linked the behavior of to questions about the Riemann zeta function. We also study the problem of evaluating Pólya's sum , and we determine some new local extrema for this function, including some new positive values.

     

Fast convolutions meet Montgomery

Arithmetic in large ring and field extensions is an important problem of symbolic computation, and it consists essentially of the combination of one multiplication and one division in the underlying ring. Methods are known for replacing one division by two short multiplications in the underlying ring, which can be performed essentially by using convolutions.

However, while using school-book multiplication, modular multiplication may be grouped into operations (where denotes the number of operations of one multiplication in the underlying ring), the short multiplication problem is an important obstruction to convolution. It raises the costs in that case to . In this paper we give a method for understanding and bypassing this problem, thus reducing the costs of ring arithmetic to roughly when also using fast convolutions. The algorithms have been implemented with results which fit well the theoretical prediction and which shall be presented in a separate paper.

     

On the absolute Mahler measure of polynomials having all zeros in a sector. II

Let be an algebraic integer of degree , not or a root of unity, all of whose conjugates are confined to a sector . In the paper On the absolute Mahler measure of polynomials having all zeros in a sector, G. Rhin and C. Smyth compute the greatest lower bound of the absolute Mahler measure ( of , for belonging to nine subintervals of . In this paper, we improve the result to thirteen subintervals of and extend some existing subintervals.

     

Elliptic curves with nonsplit mod representations

We calculate explicitly the -invariants of the elliptic curves corresponding to rational points on the modular curve by giving an expression defined over of the -function in terms of the function field generators and of the elliptic curve . As a result we exhibit infinitely many elliptic curves over with nonsplit mod representations.

     

Computing the multiplicative group of residue class rings

Let be a global field with maximal order and let be an ideal of . We present algorithms for the computation of the multiplicative group of the residue class ring and the discrete logarithm therein based on the explicit representation of the group of principal units. We show how these algorithms can be combined with other methods in order to obtain more efficient algorithms. They are applied to the computation of the ray class group modulo , where denotes a formal product of real infinite places, and also to the computation of conductors of ideal class groups and of discriminants and genera of class fields.

     

The -approximation order of surface spline interpolation

We show that if the open, bounded domain has a sufficiently smooth boundary and if the data function is sufficiently smooth, then the -norm of the error between and its surface spline interpolant is ( ), where and is an integer parameter specifying the surface spline. In case , this lower bound on the approximation order agrees with a previously obtained upper bound, and so we conclude that the -approximation order of surface spline interpolation is .

     

Short effective intervals containing primes in arithmetic progressions and the seven cubes problem

For any and any non-exceptional modulus , we prove that, for large enough ( ), the interval contains a prime in any of the arithmetic progressions modulo . We apply this result to establish that every integer larger than is a sum of seven cubes.

     

Continuous-time Kreiss resolvent condition on infinite-dimensional spaces

Given the infinitesimal generator of a -semigroup on the Banach space which satisfies the Kreiss resolvent condition, i.e., there exists an such that for all complex with positive real part, we show that for general Banach spaces this condition does not give any information on the growth of the associated -semigroup. For Hilbert spaces the situation is less dramatic. In particular, we show that the semigroup can grow at most like . Furthermore, we show that for every there exists an infinitesimal generator satisfying the Kreiss resolvent condition, but whose semigroup grows at least like . As a consequence, we find that for with the standard Euclidian norm the estimate cannot be replaced by a lower power of or .

     

Maximum norm stability of difference schemes for parabolic equations on overset nonmatching space-time grids

In this paper, theoretical results are described on the maximum norm stability and accuracy of finite difference discretizations of parabolic equations on overset nonmatching space-time grids. We consider parabolic equations containing a linear reaction term on a space-time domain which is decomposed into an overlapping collection of cylindrical subregions of the form , for . Each of the space-time domains are assumed to be independently grided (in parallel) according to the local geometry and space-time regularity of the solution, yielding space-time grids with mesh parameters and . In particular, the different space-time grids need not match on the regions of overlap, and the time steps can differ from one grid to the next. We discretize the parabolic equation on each local grid by employing an explicit or implicit -scheme in time and a finite difference scheme in space satisfying a discrete maximum principle. The local discretizations are coupled together, without the use of Lagrange multipliers, by requiring the boundary values on each space-time grid to match a suitable interpolation of the solution on adjacent grids. The resulting global discretization yields a large system of coupled equations which can be solved by a parallel Schwarz iterative procedure requiring some communication between adjacent subregions. Our analysis employs a contraction mapping argument.

Applications of the results are briefly indicated for reaction-diffusion equations with contractive terms and heterogeneous hyperbolic-parabolic approximations of parabolic equations.

     

Class numbers of real cyclotomic fields of prime conductor

The class numbers of the real cyclotomic fields are notoriously hard to compute. Indeed, the number is not known for a single prime . In this paper we present a table of the orders of certain subgroups of the class groups of the real cyclotomic fields for the primes . It is quite likely that these subgroups are in fact equal to the class groups themselves, but there is at present no hope of proving this rigorously. In the last section of the paper we argue --on the basis of the Cohen-Lenstra heuristics-- that the probability that our table is actually a table of class numbers , is at least .

     

On the computation of all extensions of a -adic field of a given degree

Let be a -adic field. It is well-known that has only finitely many extensions of a given finite degree. Krasner has given formulae for the number of extensions of a given degree and discriminant. Following his work, we present an algorithm for the computation of generating polynomials for all extensions of a given degree and discriminant.